# Regular prime

In number theory, a **regular prime** is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.

In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent *p* if *p* is regular. This focused attention on the irregular primes.^{[1]} In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent *p*, if (*p*, *p* − 3) is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either (*p*, *p* − 3) or (*p*, *p* − 5) fails to be an irregular pair.

Kummer found the irregular primes less than 165. In 1963, Lehmer reported results up to 10000 and Selfridge and Pollack announced in 1964 to have completed the table of irregular primes up to 25000. Although the two latter tables did not appear in print, Johnson found that (*p*, *p* − 3) is in fact an irregular pair for *p* = 16843 and that this is the first and only time this occurs for *p* < 30000.^{[2]} It was found in 1993 that the next time this happens is for *p* = 2124679; see Wolstenholme prime.^{[3]}

An odd prime number *p* is defined to be regular if it does not divide the class number of the *p*-th cyclotomic field **Q**(*ζ*_{p}), where *ζ*_{p} is a primitive *p*-th root of unity, it is listed on . The prime number 2 is often considered regular as well.

The class number of the cyclotomic
field is the number of ideals of the ring of integers
**Z**(ζ_{p}) up to equivalence. Two ideals *I*, *J* are considered equivalent if there is a nonzero *u* in **Q**(*ζ*_{p}) so that *I* = *uJ*.

Ernst Kummer (Kummer 1850) showed that an equivalent criterion for regularity is that *p* does not divide the numerator of any of the Bernoulli numbers *B*_{k} for *k* = 2, 4, 6, ..., *p* − 3.

Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain consequences of *p* dividing one of these Bernoulli numbers.

It has been conjectured that there are infinitely many regular primes. More precisely Carl Ludwig Siegel (1964) conjectured that *e*^{−1/2}, or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density. Neither conjecture has been proven to date.

An odd prime that is not regular is an **irregular prime** (or Bernoulli irregular or B-irregular to distinguish from other types or irregularity discussed below). The first few irregular primes are:

K. L. Jensen (an otherwise unknown student of Nielsen^{[4]}) proved in 1915 that there are infinitely many irregular primes of the form 4*n* + 3.
^{[5]}
In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.^{[6]}

Metsänkylä proved that for any integer *T* > 6, there are infinitely many irregular primes not of the form *mT* + 1 or *mT* − 1,^{[7]} and later generalized it.^{[8]}

If *p* is an irregular prime and *p* divides the numerator of the Bernoulli number *B*_{2k} for 0 < 2*k* < *p* − 1, then (*p*, 2*k*) is called an **irregular pair**. In other words, an irregular pair is a bookkeeping device to record, for an irregular prime *p*, the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs (when ordered by *k*) are:

For a given prime *p*, the number of such pairs is called the **index of irregularity** of *p*.^{[9]} Hence, a prime is regular if and only if its index of irregularity is zero. Similarly, a prime is irregular if and only if its index of irregularity is positive.

It was discovered that (*p*, *p* − 3) is in fact an irregular pair for *p* = 16843, as well as for *p* = 2124679. There are no more occurrences for *p* < 10^{9}.

An odd prime *p* has **irregular index** *n* if and only if there are *n* values of *k* for which *p* divides *B _{2k}* and these

*k*s are less than (

*p*− 1)/2. The first irregular prime with irregular index greater than 1 is 157, which divides

*B*

_{62}and

*B*

_{110}, so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0.

Similarly, we can define an **Euler irregular prime** (or E-irregular) as a prime *p* that divides at least one Euler number *E _{2n}* with 0 < 2

*n*≤

*p*− 3. The first few Euler irregular primes are

Vandiver proved that Fermat's Last Theorem (*x*^{p} + *y*^{p} = *z*^{p}) has no solution for integers *x*, *y*, *z* with gcd(*xyz*, *p*) = 1 if *p* is Euler-regular. Gut proved that *x*^{2p} + *y*^{2p} = *z*^{2p} has no solution if *p* has an E-irregularity index less than 5.^{[10]}

It was proven that there is an infinity of E-irregular primes. A stronger result was obtained: there is an infinity of E-irregular primes congruent to 1 modulo 8. As in the case of Kummer's B-regular primes, there is as yet no proof that there are infinitely many E-regular primes, though this seems likely to be true.

A prime *p* is called **strong irregular** if it's both B-irregular and E-irregular (the indexes of Bernoulli and Euler numbers that are divisible by *p* can be either the same or different). The first few strong irregular primes are

To prove the Fermat's Last Theorem for a strong irregular prime *p* is more difficult (since Kummer proved the first case of Fermat's Last Theorem for B-regular primes, Vandiver proved the first case of Fermat's Last Theorem for E-regular primes), the most difficult is that *p* is not only a strong irregular prime, but 2*p* + 1, 4*p* + 1, 8*p* + 1, 10*p* + 1, 14*p* + 1, and 16*p* + 1 are also all composite (Legendre proved the first case of Fermat's Last Theorem for primes *p* such that at least one of 2*p* + 1, 4*p* + 1, 8*p* + 1, 10*p* + 1, 14*p* + 1, and 16*p* + 1 is prime), the first few such *p* are

A prime *p* is **weak irregular** if it's either B-irregular or E-irregular (or both). The first few weak irregular primes are

Like the Bernoulli irregularity, the weak regularity relates to the divisibility of class numbers of cyclotomic fields. In fact, a prime *p* is weak irregular if and only if *p* divides the class number of the 4*p*-th cyclotomic field **Q**(*ζ*_{4p}).

In this section, "*a _{n}*" means the numerator of the

*n*th Bernoulli number if

*n*is even, "

*a*" means the (

_{n}*n*− 1)th Euler number if

*n*is odd (sequence in the OEIS).

Since for every odd prime *p*, *p* divides *a _{p}* if and only if

*p*is congruent to 1 mod 4, and since

*p*divides the denominator of (

*p*− 1)th Bernoulli number for every odd prime

*p*, so for any odd prime

*p*,

*p*cannot divide

*a*

_{p−1}. Besides, if and only if an odd prime

*p*divides

*a*(and 2

_{n}*p*does not divide

*n*), then

*p*also divides

*a*

_{n+k(p−1)}(if 2

*p*divides

*n*, then the sentence should be changed to "

*p*also divides

*a*

_{n+2kp}". In fact, if 2

*p*divides

*n*and

*p*(

*p*− 1) does not divide

*n*, then

*p*divides

*a*

_{n}.) for every integer

*k*(a condition is

*n*+

*k*(

*p*− 1) must be > 1). For example, since 19 divides

*a*

_{11}and 2 × 19 = 38 does not divide 11, so 19 divides

*a*

_{18k+11}for all

*k*. Thus, the definition of irregular pair (

*p*,

*n*),

*n*should be at most

*p*− 2.

The only primes below 1000 with weak irregular index 3 are 307, 311, 353, 379, 577, 587, 617, 619, 647, 691, 751, and 929. Besides, 491 is the only prime below 1000 with weak irregular index 4, and all other odd primes below 1000 with weak irregular index 0, 1, or 2. (**Weak irregular index** is defined as "number of integers 0 ≤ *n* ≤ *p* − 2 such that *p* divides *a _{n}*.)

The following table shows all irregular pairs with *n* ≤ 63. (To get these irregular pairs, we only need to factorize *a _{n}*. For example,

*a*

_{34}= 17 × 151628697551, but 17 < 34 + 2, so the only irregular pair with

*n*= 34 is (151628697551, 34)) (for more information (even

*n*s up to 300 and odd

*n*s up to 201), see

^{[11]}).

The following table shows irregular pairs (*p*, *p* − *n*) (*n* ≥ 2), it is a conjecture that there are infinitely many irregular pairs (*p*, *p* − *n*) for every natural number *n* ≥ 2, but only few were found for fixed *n*. For some values of *n*, even there is no known such prime *p*.